A008284 -id:A008284 - cp's OEIS frontend

A058398 Partition triangle A008284 read from right to left.

1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 2, 2, 1, 1, 1, 2, 3, 3, 1, 1, 1, 2, 3, 4, 3, 1, 1, 1, 2, 3, 5, 5, 4, 1, 1, 1, 2, 3, 5, 6, 7, 4, 1, 1, 1, 2, 3, 5, 7, 9, 8, 5, 1, 1, 1, 2, 3, 5, 7, 10, 11, 10, 5, 1, 1, 1, 2, 3, 5, 7, 11, 13, 15, 12, 6, 1, 1, 1, 2, 3, 5, 7, 11, 14, 18, 18, 14, 6, 1, 1, 1, 2, 3, 5, 7, 11

Offset: 1

Wolfdieter Lang, Dec 11 2000

a(n,m) is the number of partitions of n with n-(m-1) parts or, equivalently, with greatest part n-(m-1).

The columns are the diagonals of triangle A008284. The diagonals are the columns of the partition array p(n,m), n >= 0, m >= 1, with p(n,m) the number of partitions of n in which every part is <= m; p(0,m) := 1. For n >= 1 this array is obtained from table A026820 read as lower triangular array with extension of the rows according to p(n,m)=A000041(n) for m>n.

			Lower triangular matrix:
  1;
  1,1;
  1,1,1;
  1,1,2,1;
  1,1,2,2,1;
  1,1,2,3,3,1;
  1,1,2,3,4,3,1;
  1,1,2,3,5,5,4,1;
  ...

L. Comtet, Advanced Combinatorics, Reidel, 1974, pp. 94, 96 and 307.
M. Kauers and P. Paule, The Concrete Tetrahedron, Springer 2011, p. 27.

Seiichi Manyama, Rows n = 1..140, flattened
Henry Bottomley, Illustration of initial terms
Roser Homs and Anna-Lena Winz, Deformations of local Artin rings via Hilbert-Burch matrices, arXiv:2309.06871 [math.AC], 2023. See p. 16.

Cf. A026820, A008284, A000041.

row[n_] := Table[ IntegerPartitions[n, k] // Length, {k, 0, n}] // Differences // Reverse; Table[row[n], {n, 1, 14}] // Flatten (* Jean-François Alcover, Feb 28 2013 *)

a(n, m)= A008284(n, n-(m-1)).
a(n, m)= p(m-1, n-m+1), n >= m >= 1 with the p(n, m) array defined in the comment.
a(n, m)=0 if nViewed as a square array by antidiagonals, T(n,k) = 0 if n<0; T(n,1) = 1; otherwise T(n,k) = T(n,k-1) + T(n-k,k). - Franklin T. Adams-Watters, Jul 25 2006
Let x be a triangular number C(n,2), where n is the integer being partitioned.  Then a(x) = a(x+1) = a(x+2) = 1.  Also, a(x+3) = 2 for x>3 and a(x-1) = floor(n/2). - Allan Bickle, Apr 18 2024

A038498 Matrix inverse of partition triangle A008284.

Original entry on oeis.org

1, -1, 1, 0, -1, 1, 1, -1, -1, 1, 0, 1, -1, -1, 1, 0, 1, 0, -1, -1, 1, -1, 1, 1, 0, -1, -1, 1, -1, 0, 2, 0, 0, -1, -1, 1, 0, -1, 0, 2, 0, 0, -1, -1, 1, 0, -2, 1, 1, 1, 0, 0, -1, -1, 1, 1, -2, -1, 1, 1, 1, 0, 0, -1, -1, 1, 1, -1, -2, 0, 2, 0, 1, 0, 0, -1, -1, 1

Offset: 1

Christian G. Bower, Feb 15 1999

Since A008284 has only ones in its first column, the sum of terms for any row n > 1 is 0. - François Marques, Feb 09 2021

			Triangle begins:
  1;
  -1,1;
  0,-1,1;
  1,-1,-1,1;
  ...

Cf. A008284, A038497, A039800-A039810, A010815.

tp(n, k) = if (n<1, 0, if (k<1, 0, if (k == n, 1, if (k > n, 0, tp(n-1, k-1) + tp(n-k, k)))));
tabl(nn) = {mtp = matrix(nn, nn, n, k, tp(n, k)); mtpi = mtp^(-1); for (n = 1, nn, for (k = 1, n, print1(mtpi[n, k], ", ");); print(););} \\ Michel Marcus, Mar 04 2014

T(n,n-k) = A010815(k) for k <= n/2. - François Marques, Feb 09 2021

A055884 Euler transform of partition triangle A008284.

Original entry on oeis.org

1, 1, 2, 1, 2, 3, 1, 4, 4, 5, 1, 4, 8, 7, 7, 1, 6, 12, 16, 12, 11, 1, 6, 17, 25, 28, 19, 15, 1, 8, 22, 43, 49, 48, 30, 22, 1, 8, 30, 58, 87, 88, 77, 45, 30, 1, 10, 36, 87, 134, 167, 151, 122, 67, 42, 1, 10, 45, 113, 207, 270, 296, 247, 185, 97, 56, 1, 12, 54, 155, 295, 448, 510, 507, 394, 278, 139, 77

Offset: 1

Christian G. Bower, Jun 09 2000

Number of multiset partitions of length-k integer partitions of n. - Gus Wiseman, Nov 09 2018

			From _Gus Wiseman_, Nov 09 2018: (Start)
Triangle begins:
   1
   1   2
   1   2   3
   1   4   4   5
   1   4   8   7   7
   1   6  12  16  12  11
   1   6  17  25  28  19  15
   1   8  22  43  49  48  30  22
   1   8  30  58  87  88  77  45  30
   ...
The fifth row {1, 4, 8, 7, 7} counts the following multiset partitions:
  {{5}}   {{1,4}}     {{1,1,3}}       {{1,1,1,2}}         {{1,1,1,1,1}}
          {{2,3}}     {{1,2,2}}      {{1},{1,1,2}}       {{1},{1,1,1,1}}
         {{1},{4}}   {{1},{1,3}}     {{1,1},{1,2}}       {{1,1},{1,1,1}}
         {{2},{3}}   {{1},{2,2}}     {{2},{1,1,1}}      {{1},{1},{1,1,1}}
                     {{2},{1,2}}    {{1},{1},{1,2}}     {{1},{1,1},{1,1}}
                     {{3},{1,1}}    {{1},{2},{1,1}}    {{1},{1},{1},{1,1}}
                    {{1},{1},{3}}  {{1},{1},{1},{2}}  {{1},{1},{1},{1},{1}}
                    {{1},{2},{2}}
(End)

Alois P. Heinz, Rows n = 1..200, flattened
N. J. A. Sloane, Transforms

Row sums give A001970.
Main diagonal gives A000041.
Columns k=1-2 give: A057427, A052928.
T(n+2,n+1) gives A000070.
T(2n,n) gives A360468.
Cf. A055885, A055886, A360742.
Cf. A000219, A007716, A008284, A255906, A317449, A317532, A317533, A320796, A320801, A320808.

h:= proc(n, i) option remember; expand(`if`(n=0, 1,
      `if`(i<1, 0, h(n, i-1)+x*h(n-i, min(n-i, i)))))
    end:
g:= proc(n, i, j) option remember; expand(`if`(j=0, 1, `if`(i<0, 0, add(
      g(n, i-1, j-k)*x^(i*k)*binomial(coeff(h(n$2), x, i)+k-1, k), k=0..j))))
    end:
b:= proc(n, i) option remember; expand(`if`(n=0, 1,
     `if`(i<1, 0, add(b(n-i*j, i-1)*g(i$2, j), j=0..n/i))))
    end:
T:= (n, k)-> coeff(b(n$2), x, k):
seq(seq(T(n,k), k=1..n), n=1..12);  # Alois P. Heinz, Feb 17 2023

sps[{}]:={{}};sps[set:{i_,_}]:=Join@@Function[s,Prepend[#,s]&/@sps[Complement[set,s]]]/@Cases[Subsets[set],{i,_}];
mps[set_]:=Union[Sort[Sort/@(#/.x_Integer:>set[[x]])]&/@sps[Range[Length[set]]]];
Table[Length[Join@@mps/@IntegerPartitions[n,{k}]],{n,5},{k,n}] (* Gus Wiseman, Nov 09 2018 *)

A058399 Triangle of partial row sums of partition triangle A008284.

Original entry on oeis.org

1, 2, 1, 3, 2, 1, 5, 4, 2, 1, 7, 6, 4, 2, 1, 11, 10, 7, 4, 2, 1, 15, 14, 11, 7, 4, 2, 1, 22, 21, 17, 12, 7, 4, 2, 1, 30, 29, 25, 18, 12, 7, 4, 2, 1, 42, 41, 36, 28, 19, 12, 7, 4, 2, 1, 56, 55, 50, 40, 29, 19, 12, 7, 4, 2, 1, 77, 76, 70, 58, 43, 30, 19, 12, 7, 4, 2, 1, 101, 100, 94, 80, 62

Offset: 1

Wolfdieter Lang, Dec 11 2000

T(n,m) is also the number of m-th largest elements in all partitions of n. - Omar E. Pol, Feb 14 2012
It appears that reversed rows converge to A000070. - Omar E. Pol, Mar 10 2012
The row sums give A006128. - Omar E. Pol, Mar 26 2012
T(n,m) is also the number of regions traversed by the m-th column of the section model of partitions with n sections (Cf. A135010, A206437). - Omar E. Pol, Apr 20 2012

			From _Omar E. Pol_, Mar 10 2012: (Start)
Triangle begins:
   1;
   2,  1;
   3,  2,  1;
   5,  4,  2,  1;
   7,  6,  4,  2,  1;
  11, 10,  7,  4,  2,  1;
  15, 14, 11,  7,  4,  2,  1;
  22, 21, 17, 12,  7,  4,  2,  1;
  30, 29, 25, 18, 12,  7,  4,  2,  1;
  42, 41, 36, 28, 19, 12,  7,  4,  2,  1;
  56, 55, 50, 40, 29, 19, 12,  7,  4,  2,  1;
  77, 76, 70, 58, 43, 30, 19, 12,  7,  4,  2,  1;
(End)

Alois P. Heinz, Rows n = 1..141, flattened

Columns 1-5: A000041(n), A000065(n+1), A004250(n+2), A035300(n-1), A035301(n-1), n >= 1.

Cf. A008284.

b:= proc(n, k) option remember;
      `if`(n=0, 1, `if`(k<1, 0, add(b(n-j*k, k-1), j=0..n/k)))
    end:
T:= (n, m)-> b(n,n) -b(n,m-1):
seq (seq (T(n, m), m=1..n), n=1..15);  # Alois P. Heinz, Apr 20 2012

t[n_, m_] := Sum[ IntegerPartitions[n, {k}] // Length, {k, m, n}]; Table[t[n, m], {n, 1, 13}, {m, 1, n}] // Flatten (* Jean-François Alcover, Jun 21 2013 *)

T(n, m) = Sum_{k=m..n} A008284(n, k).
G.f. for m-th column: Sum_{n>=1} x^(n)/Product_{k=1..n+m-1} (1 - x^k).
T(n, m) = Sum_{k=1..n} A207379(k, m). - Omar E. Pol, Apr 22 2012

A038497 Matrix square of partition triangle A008284.

Original entry on oeis.org

1, 2, 1, 3, 2, 1, 5, 5, 2, 1, 7, 8, 5, 2, 1, 11, 15, 10, 5, 2, 1, 15, 23, 18, 10, 5, 2, 1, 22, 38, 31, 20, 10, 5, 2, 1, 30, 56, 52, 34, 20, 10, 5, 2, 1, 42, 86, 83, 60, 36, 20, 10, 5, 2, 1, 56, 123, 129, 97, 63, 36, 20, 10, 5, 2, 1, 77, 181, 198, 158, 105, 65, 36, 20, 10, 5, 2, 1

Offset: 1

Christian G. Bower, Feb 15 1999

Row sums form A022811 (number of terms in n-th derivative of a function composed with itself 3 times). - Paul D. Hanna, Jul 13 2004

			1; 2,1; 3,2,1; 5,5,2,1; ...

Cf. A038498, A039800-A039809. a(n, 1) = A000041(n) (first column) (partition numbers).

Cf. A022811.

A039805 Matrix cube of partition triangle A008284.

Original entry on oeis.org

1, 3, 1, 6, 3, 1, 13, 9, 3, 1, 23, 19, 9, 3, 1, 44, 42, 22, 9, 3, 1, 74, 80, 48, 22, 9, 3, 1, 129, 154, 99, 51, 22, 9, 3, 1, 210, 273, 193, 105, 51, 22, 9, 3, 1, 345, 484, 362, 212, 108, 51, 22, 9, 3, 1, 542, 815, 651, 401, 218, 108, 51, 22, 9, 3, 1, 858, 1369, 1147

Offset: 1

Christian G. Bower, Feb 15 1999

Row sums form A022812 (number of terms in n-th derivative of a function composed with itself 4 times). - Paul D. Hanna, Jul 13 2004

			1; 3,1; 6,3,1; 13,9,3,1; ...

Cf. A038497, A038498, A039806, A039807. a(n, 1) = A022811(n) (first column).

Cf. A022812.

A327029 T(n, k) = Sum_{d|n} phi(d) * A008284(n/d, k) for n >= 1, T(0, 0) = 1. Triangle read by rows for 0 <= k <= n.

Original entry on oeis.org

1, 0, 1, 0, 2, 1, 0, 3, 1, 1, 0, 4, 3, 1, 1, 0, 5, 2, 2, 1, 1, 0, 6, 6, 4, 2, 1, 1, 0, 7, 3, 4, 3, 2, 1, 1, 0, 8, 8, 6, 6, 3, 2, 1, 1, 0, 9, 6, 9, 6, 5, 3, 2, 1, 1, 0, 10, 11, 10, 10, 8, 5, 3, 2, 1, 1, 0, 11, 5, 10, 11, 10, 7, 5, 3, 2, 1, 1, 0, 12, 17, 19, 19, 14, 12, 7, 5, 3, 2, 1, 1

Offset: 0

Peter Luschny, Aug 24 2019

Dirichlet convolution of phi(n) and A008284(n,k) for n >= 1. - Richard L. Ollerton, May 07 2021

			Triangle starts:
[0] [1]
[1] [0, 1]
[2] [0, 2, 1]
[3] [0, 3, 1, 1]
[4] [0, 4, 3, 1, 1]
[5] [0, 5, 2, 2, 1, 1]
[6] [0, 6, 6, 4, 2, 1, 1]
[7] [0, 7, 3, 4, 3, 2, 1, 1]
[8] [0, 8, 8, 6, 6, 3, 2, 1, 1]
[9] [0, 9, 6, 9, 6, 5, 3, 2, 1, 1]

Alois P. Heinz, Rows n = 0..200, flattened

Cf. A008284, A000010, A078392 (row sums), A282750.
Cf. A000041 (where reversed rows converge to).
T(2n,n) gives A052810.

def DivisorTriangle(f, T, Len, w = None):
    D = [[1]]
    for n in (1..Len-1):
        r = lambda k: [f(d)*T(n//d,k) for d in divisors(n)]
        L = [sum(r(k)) for k in (0..n)]
        if w != None: L = [*map(lambda v: v * w(n), L)]
        D.append(L)
    return D
DivisorTriangle(euler_phi, A008284, 10)

From Richard L. Ollerton, May 07 2021: (Start)
For n >= 1, T(n,k) = Sum_{i=1..n} A008284(gcd(n,i),k).
For n >= 1, T(n,k) = Sum_{i=1..n} A008284(n/gcd(n,i),k)*phi(gcd(n,i))/phi(n/gcd(n,i)). (End)

A039807 Matrix 5th power of partition triangle A008284.

Original entry on oeis.org

1, 5, 1, 15, 5, 1, 45, 20, 5, 1, 110, 60, 20, 5, 1, 271, 170, 65, 20, 5, 1, 599, 426, 185, 65, 20, 5, 1, 1309, 1025, 486, 190, 65, 20, 5, 1, 2690, 2299, 1185, 501, 190, 65, 20, 5, 1, 5436, 4999, 2750, 1245, 506, 190, 65, 20, 5, 1, 10545, 10380, 6069, 2910, 1260

Offset: 0

Christian G. Bower, Feb 15 1999

			1; 5,1; 15,5,1; 45,20,5,1; ...

Cf. A038497, A038498, A039805, A039806. a(n, 1) = A022813(n) (first column).

A050304 Matrix 10th power of partition triangle A008284.

Original entry on oeis.org

1, 10, 1, 55, 10, 1, 265, 65, 10, 1, 1045, 320, 65, 10, 1, 3817, 1365, 330, 65, 10, 1, 12583, 5127, 1420, 330, 65, 10, 1, 39148, 17710, 5447, 1430, 330, 65, 10, 1, 114235, 56638, 19030, 5502, 1430, 330, 65, 10, 1, 318857, 170993, 61865, 19350, 5512

Offset: 1

Christian G. Bower, Aug 15 1999

			1; 10,1; 55,10,1; 265,65,10,1; ...

Cf. A038497, A038498, A039805-A039807. A050300-A050303. a(n, 1) = A024210(n) (first column).

A039809 For n > 1, a(n) doubles under the transform T, where Ta is the matrix product of partition triangle A008284 with a, with a(1) = 1.

Original entry on oeis.org

1, 1, 2, 5, 12, 32, 83, 223, 594, 1600, 4297, 11589, 31216, 84212, 227091, 612712, 1652913, 4459962, 12033405, 32469682, 87611105, 236402465, 637884103, 1721218224, 4644392797, 12532091909, 33815653370, 91245738923

Offset: 1

Christian G. Bower, Feb 15 1999

			So a(7) = T(7,1)*a(1) + T(7,2)*a(2) + ... + T(7,6)*a(6) = 1*1 + 3*1 + 4*2 + 3*5 + 2*12 + 1*32 = 1 + 3 + 8 + 15 + 24 + 32 = 83, where T(n,k) = A008284(n,k).

Cf. A008284, A039808, A081719.

P(n, k) = #partitions(n-k, k); /* A008284 */
lista(nn) = {my(a=vector(nn)); a[1]=1; for(n=2, nn, a[n] = sum(i=1, n-1, P(n,i)*a[i])); a;} \\ Petros Hadjicostas, May 30 2020

a(1) = 1 and a(n) = Sum_{i=1..n-1} A008284(n, i)*a(i) for n >= 2 (because 2*a(n) = Sum_{i=1..n} A008284(n,i)*a(i) for n >= 2).
a(n+1) = Sum_{k=0..n} A081719(n,k). - Philippe Deléham, Sep 30 2006
G.f.: (1/2) * ( x + Sum_{n>=1} a(n) * x^n / Product_{j=1..n} (1 - x^j) ). - Ilya Gutkovskiy, Jul 22 2021

Various sections edited by Petros Hadjicostas, May 30 2020

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cp's OEIS Frontend

A058398 Partition triangle A008284 read from right to left.

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A038498 Matrix inverse of partition triangle A008284.

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A055884 Euler transform of partition triangle A008284.

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A058399 Triangle of partial row sums of partition triangle A008284.

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Maple

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A038497 Matrix square of partition triangle A008284.

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A039805 Matrix cube of partition triangle A008284.

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A327029 T(n, k) = Sum_{d|n} phi(d) * A008284(n/d, k) for n >= 1, T(0, 0) = 1. Triangle read by rows for 0 <= k <= n.

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A039807 Matrix 5th power of partition triangle A008284.

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A050304 Matrix 10th power of partition triangle A008284.

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